LOGIN TO YOUR ACCOUNT

Username
Password
Remember Me
Or use your Academic/Social account:

CREATE AN ACCOUNT

Or use your Academic/Social account:

Congratulations!

You have just completed your registration at OpenAire.

Before you can login to the site, you will need to activate your account. An e-mail will be sent to you with the proper instructions.

Important!

Please note that this site is currently undergoing Beta testing.
Any new content you create is not guaranteed to be present to the final version of the site upon release.

Thank you for your patience,
OpenAire Dev Team.

Close This Message

CREATE AN ACCOUNT

Name:
Username:
Password:
Verify Password:
E-mail:
Verify E-mail:
*All Fields Are Required.
Please Verify You Are Human:
fbtwitterlinkedinvimeoflicker grey 14rssslideshare1
Hunt, John H. V.,1940-
Languages: English
Types: Doctoral thesis
Subjects: QA
We have mentioned that each chapter in this thesis is conceived of as an independent paper, except for Chapter 3, which is a collection of results on non- continuous functions. Consequently each chapter contains a clearly marked introductory section, in which its back- ground and content are explained. In this abstract we shall summarize the remarks in these introductory sections.\ud \ud In chapter 1 we present an n-arc theorem for Peano spaces which is an extension of the theorem in §2 of [32], which Menger called the second n-arc theorem in [17]. Whereas in the second n-arc theorem n disjoint arcs are constructed joining two disjoint closed sets A and B, in chapter 1 we split the closed set A into n dis- joint closed subsets A1 A 2, ••• , An and give necessary and sufficient conditions for there to be n disjoint arcs joining A and B, one meeting each A1. At the end of chapter 1 we present a conjecture, which we have been able to verify in special cases.\ud \ud In [35] Whyburn proved a theorem concerning the weak connected separation of two non-degenerate connected closed sets A and B by a quasi-closed set L in a locally cohesive space X. In chapter 2 we show that A and B can in fact be taken as arbitrary closed sets in this theorem; that is, ,we remove the restriction of non-degeneracy and connectedness on A and B.\ud \ud In chapter 3 we study the circumstances under which a connectivity function is peripherally continuous. \ud \ud The study of the abstract relations between non- continuous functions was initiated by Stallings in [23]. In this paper he introduced the 1pc polyhedron and showed that a connectivity function was peripherally continuous on an 1pc polyhedron. Whyburn took up the study of non- continuous functions in [33]. [34] and [35]. He introduced the locally cohesive space, which is more general than the 1pc polyhedron, and proved that a connectivity function was peripherally continuous on a locally cohesive Peano space.\ud \ud For technical reasons, the locally cohesive space is not permitted to have local cut points. It is obvious, however, that on many Peano spaces having local, cut points a connectivity function remains peripherally continuous, In §2,3 of chapter 3 we formulate a sequence of properties Pn(X), which permit the space X to have local cut points, and we prove in each case that a connectivity function f : X →Y is peripherally continuous when X has property Pn(X). Each of these properties is an improvement on the last, and the final one, the U-space, satisfactorily incorporates the class of Peano spaces with local cut points on which we are able to prove that a connectivity function is peripherally continuous.\ud \ud An interesting feature of §3 of chapter 3 is provided by two "weak separation theorems," and more will be found about these in the introduction to chapter 3.\ud \ud In §4 of chapter 3 we show that a connectivity function is peripherally continuous on a locally compact ANR. This affirmatively answers a question that Stallings raised in [23].\ud \ud The U-space that we have introduced in §3 of chapter 3 imposes a "unicoherence condition" in the space X (as do all the properties Pn(X) considered in §3, chapter 3). In §5 of chapter 3 we generalize the U-space to the S-space. This imposes a "multicoherence condition" on the space X, and we prove that a connectivity function is peripherally continuous on a cyclic S-space.\ud \ud We close chapter 3 by considering the question of placing weaker conditions than connectivity on the function f : X → Y which will still ensure that f is peripherally continuous. \ud \ud It is well known that if X is a unicoherent Peano continuum and A1, A 2, … is a sequence of disjoint closed subsets of X no one of which separates X, then Un=1 An does not separate X. In [28] van Est proved this theorem for the case where X is a Euclidean space of n dimensions. In chapter 4 we give an example which shows that this theorem does not hold if X is an arbitrary Peano space •\ud \ud In chapter 5 we provide a new angle to Lebesgue's covering lemma. We show that if the Lebesgue number ᵹ of an open covering U1, U2, •••• Un of a compact metric space X. ρ is finite. then it can be defined by the formula ᵹ = min ρ (E, F), where E and F are any compartments contained in no common U1\ud \ud In chapter 6 we show that an involution on a cyclic Peano space leaves some simple closed curve setwise invariant. \ud \ud Whyburn has given a proof of R. L. Moore’s decomposition theorem for the 2-sphere in [31] (a refinement of this proof is presented in [36]). His proof is accomplished by showing that the decomposition space satisfies Zippin’s characterization theorem for the 2-sphere. In chapter 6 we present an alternative way of showing that the decomposition space satisfies Zippin's characterization theorem. Our proof closely follows Alexander's proof of the Jordan curve theorem as given by Newman in[21], and so consists of arguments that are well-known in another context.\ud \ud In [30] Whyburn gave a proof of the cyclic connectivity theorem. and in all subsequent appearances of this theorem in the literature Whyburn's proof has been used. Whyburn divided the proof of the theorem into three parts lemma 1, lemma 2, and the deduction of the theorem from lemmas 1 and2. In chapter 8 we give an alternative proof of lemma 1. Our proof is based on the fact that a cyclic Peano space has a base of regions whose closures do not separate the space, and it proceeds by an induction on a simple chain of these regions.\ud
  • The results below are discovered through our pilot algorithms. Let us know how we are doing!

    • 6. J. L. Cornette, Connectivity functions and i~a5es on Peano continua, Fund. Math. 58 (1966) pp. 183-192.
    • 7. J. L. Cornette and J. Z. Girolo, Connectivity retracts of finitely coherent Peano continua, Fund. Math. 61 (1967) pp. 177-182.
    • 8. A. Garc{a Naynez, Concerning partially continuous functions, Ph. D. thesis, University of Virginia, Charlottesville, 1968.
    • 9. Helvin R. Hagan, Upper semi-continuous decompositions and factorization of certain non-continuous transformations, Duke Math. J. 32 (1965) pp. 679-688.
    • 10. - - - , Equivalence of connectivi ty m2.D8 and peripherally continuous transformations, Proc . .AtU~ller. ~ath. Soc. 17 (1966) Pp. 175-177.
    • 11. D. Wo Hall and G. L. Spencer, Elementary topology, John Wiley & Sons, New York, 1955.
    • 12. o. H. Hamilton, Fixed points for certain noncontinuous transformations, Proc. Amer. ~lath. Soc. 8 (1957) pp. 750-756.
    • 13. F. Hausdorff, Set theory, 2nd ed., Chelsea Publishing Co., New York, 1962.
    • 14. C. Kuratot-lski, Topologie, Vol. II, 3rd ed., PWN, Warsaw, 1961.
    • 15. - - - , Introduction to set theory and tODology, International series of monographs on pure and applied mathematics, Vol. 13, Pergamon Press, Oxford, 1961.
    • 16. S. Mazurkiewicz, Remargue sur un the/ oreme de Iii. rvrullikin, Fund. Math. 6 (1924) pp. 37-38.
    • 17. K. Menger, Kurventheorie,. Teubner, BerlinLeipzig, 1932, chap. VI.
    • 18. R. L. Moore, On the foundations of plane analysis situs, Trans. Amer. I'1ath. Soc. 17 (1916) pp. 131-164.
    • 190 ________ , Concerning UDDer semi-continuous collections of continua, Trans. Amer. Math. Soc. 27 (1925) pp. 416-428.
    • 20. A. M. ~ullikin, Certain theorems relatin~ ~o DIane connected point sets, Trans. Amer. Nath. Soc. 24 (1923) pp. 144-162.
    • 21. r,l. H. A. Newman, Elements of the topology of .?18.ne sets of pOints, 2nd ed., Cambridge University Press, Cambridge, 1951.
    • 22. D. E. Sanderson, Relations among some basic properties of non-continuous functions, Duke Math. J. 35 (1968) pp. 407-414.
    • 23. J. Stallings, Fixed point theorems for connectivity maps, Fund. ~1ath. 47 (1959) pp. 249-263.
    • 24. A. H. Stone, Incidence relations in unicoheren~ spaces, Trans. Amer. Hath. Soc. 65 (1949) pp. 427-447.
    • 25. ______ , Incidence relations in multicoherent spaces I, Trans. Amer. Math. Soc. 66 (1949) pp. 389-406.
    • 26. ------, On infinitely multicoherent spaces, Quart. J. Ivlath. Oxford (2), 3 (1952) pp. 298-306.
    • 27. E. S. Thomas, Jr., Almost continuous functions and functions of Baire class 1, Annals of Mathematics Studies No. 60 (Topology Seminar, Wisconsin, 1965, edited by R. H. Bing and R. J. Bean), Princeton University Press, Princeton, 1966.
    • 28. W. T. van Est, A generalization of a theorem of russ Anna Mullikin, Fund. Math. 39 (1952) pp. 179-188.
    • 29. Eo R. van Kampen, On some characteriz8tions of 2-dimensional manifolds, Duke Math. J. 1 (1935) pp. 74-93.
    • 30. G. T. Whyburn, On the cyclic connectivitv " theorem, Bull. Amer. Math. Soc. 37 (1931) pp. 429-433.
    • 31. - - - , Analytic Tonolo~y, Amer. l'Iath. Soc. Colloquium Publications, vol. 28. 194·2.
    • 32. -----, On n-arc connectedness, Trans. Amer. Math. Soc. 63 (1948) pp. 452-456.
    • 33. ----, Connectivity of perinherally continuous functions, Proc. Nat. Acad. Sci. U.S.A. 55 (1966) pp. 1040-1041.
    • 34. ------, Loosely closed sets and partially continuous functions, Michigan Hath. J. 14 (1967) pp. 193-205.
    • 35. ______ , Quasi-closed sets and fixed points, Proc. Nat. Acad. Sci. U.S.A. 57 (1967) pp. 201-205.
    • 36. G. T. Whyburn, assisted by R. F. Dickman, ~otes on general topology and mapping theory, University of Virginia, Charlottesville, 1964-65 (mimeographed notes).
    • 37. G. T. Whyburn, assisted by J. 'H. V. Hunt, Notes on functions and multifunctions, University of Virginia, Charlottesville, 1966-67 (mimeographed notes).
    • 38. R. L. ~Hlder, Tonology of flanifolds, Amer. i'IIath. Soc. Colloquium Publications, vol. 32, 1949. . Locally contrac ti ble, 89
  • No related research data.
  • No similar publications.

Share - Bookmark

Cite this article